Explanation Calculation: Show the free body diagram of the beam as in Figure 1. The beam and loading is symmetric. R C = R A Calculate the force ( F ) in the spring: F = 1 3 ( w ) ( 2 L ) = 2 3 w L Calculate the vertical reaction ( R A ) at point A by resolving the vertical component force. ∑ F y = 0 R A + F − w × ( 2 L ) + R C = 0 2 R A + ( 2 3 w L ) − w × ( 2 L ) = 0 R A = ( 4 3 w L ) 2 R A = R C = 2 3 w L Calculate the moment due to the reaction at A : M 1 = R A × L A B Here, L A B is length of the beam. Substitute 2 3 w L for R A and L for L A B . M 1 = 2 3 w L × L = 2 3 w L 2 Calculate the M 1 E I value: Substitute 2 3 w L 2 for M 1 . ( M 1 E I ) = 2 3 w L 2 E I = 2 w L 2 3 E I Show the ( M 1 E I ) diagram in below Figure 2. Calculate the area ( A 1 ) using the relation: A 1 = 1 2 ( L A B ) ( M 1 E I ) Substitute L for L A B and 2 w L 2 3 E I for ( M 1 E I ) . A 1 = 1 2 ( L ) ( 2 w L 2 3 E I ) = w L 3 3 E I Calculate the moment ( M 2 ) due the UDL: M 2 = − w × L × L 2 = − w L 2 2 Calculate the ( M 2 E I ) value: Substitute − w L 2 2 for M 2 . ( M 2 E I ) = − w L 2 2 E I = − w L 2 2 E I Show the ( M 2 E I ) diagram in below Figure 3

7th Edition

ISBN: 9780073398235

Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek

Publisher: McGraw-Hill Education

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Chapter 9.6, Problem 155P

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The spring constant (k) for which the force in the spring is equal to one third of the total load on the beam.

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Chapter 9.6, Problem 154P

Chapter 9.6, Problem 156P

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The spring constant ( k ) for which the force in the spring is equal to one third of the total load on the beam.

Solution Summary: The author shows the free body diagram of the beam as in Figure 1. The beam and loading is symmetric.

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The spring constant (k) for which the force in the spring is equal to one third of the total load on the beam.

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